Security officials and media have reported a severe deterioration in the 84-year-old's health since he was sentenced to life last week for his role in the killing of hundreds of protesters during last year's popular uprising that toppled him.
"There are around 200 Mubarak's supporters and lawyers in front of the Prisons Authority and we will not leave until Mubarak is transferred to either a military or private hospital," said Mohamed Abdel Razek, one of Mubarak's lawyers.
Abdel Razek said the general prosecutor had told him there was "no barrier" to his request to move Mubarak from the hospital wing of Cairo's Tora prison but that the decision had to be taken by the Prisons Authority.
Last week, security sources said Mubarak was given artificial respiration five times in one day and doctors recommended he be moved to a military hospital or back to the medical facility he was in prior to his conviction.
State's news agency MENA said on Saturday Mubarak - who was wheeled into court to attend his trial on a hospital bed - risked a stroke.
The acquittal of six of Mubarak's top security officials angered those who say his old guard is still ruling from behind the scenes.
Some Egyptians had wanted Mubarak executed and the sentencing triggered days of nationwide protests demanding both a retrial and enforcement of a law, passed by parliament but not implemented, banning Mubarak-era officials from politics. The constitutional court will rule on the law's validity on June 14.
A June 16-17 presidential run-off election will pit Mubarak's last prime minister Ahmed Shafik against Muslim Brotherhood candidate Mohamed Mursi, the final step before the army, which took charge when Mubarak was toppled, formally hands over power by July 1.